The $L_p$ Gauss dual Minkowski problem

This article introduces the $L_p$-Gauss dual curvature measure and proposes its related $L_p$-Gauss dual Minkowski problem as: for $p,q\in\mathbb{R}$, under what necessary and/or sufficient condition on a non-zero finite Borel measure $\mu$ on unit sphere does there exist a convex body $K$ such that $\mu$ is the $L_p$ Gauss dual curvature measure? If $K$ exists, to what extent is it unique? This problem amounts to solving a class of Monge-Amp\`{e}re type equations on unit sphere in smooth case: \begin{align} e^{-\frac{|\nabla h_K|^2+h_K^2}{2}}h_K^{1-p} (|\nabla h_K|^2+h_K^2)^{\frac{q-n}{2}} \det(\nabla^2h_K+h_KI)=f,\qquad (0.1) \end{align} where $f$ is a given positive smooth function on unit sphere, $h_k$ is the support function of convex body $K$, $\nabla h_K$ and $\nabla^2h_K$ are the gradient and Hessian of $h_K$ on unit sphere with respect to an orthonormal basis, and $I$ is the identity matrix. We confirm the existence of solution to the new problem with $p,q>0$ and the existence of smooth solution to the equation (0.1) with $p ,q\in\mathbb{R}$ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case $p,q\in\mathbb{R}$ with $q<p$ is established.